Arithmetic Properties of Certain Level One Mock Modular Forms

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS

MATTHEW BOYLAN

Abstract. In recent work, Bringmann and Ono [4] show that Ramanujan’s f(q) mock theta function is the holomorphic projection of a harmonic weak Maass form of weight 1/2. In this paper, we extend work of Ono in [13]. In particular, we study holomorphic projections of certain integer weight harmonic weak Maass forms on SL2(Z) using Hecke operators and the diﬀerential theta-operator.

1. Introduction and statement of results.

If k is an integer, we denote by Mk (respectively, Sk) the space of holomorphic modular forms (respectively, cusp forms) of weight k on SL2(Z). Let z h, the complex upper half- 2πiz ∈ j plane, and let q := e . For integers n 1 and j 0, define σj(n) := d n d , and let Bj ≥ ≥ | denote the jth Bernoulli number. Then the Delta-function and the normalized Eisenstein series for SL (Z) of even weight k 2 are given by P 2 ≥ ∞ ∞ (1.1) ∆(z)= τ(n)qn = q (1 qm)24 S , − ∈ 12 n=1 m=1 X Y ∞ 2k n (1.2) Ek(z)=1 σk 1(n)q . − B − k n=1 X If k 4, then Ek(z) Mk. We also set E0(z) := 1. If the dimension of Sk is one, then k ≥12, 16, 18, 20, 22, 26∈ . We note that the corresponding weight k cusp forms, ∈{ } ∞ n (1.3) fk(z) := af (n)q = ∆(z)Ek 12(z), k − n=1 X have integer coefficients. In [13], Ono defined a harmonic weak Maass form related to the Delta-function and studied its holomorphic projection. In this paper, we extend Ono’s work by defining harmonic weak Maass forms related to the forms fk(z) and by studying their holomorphic projections. We denote spaces of harmonic weak Maass forms of weight j on SL2(Z) by Hj. In Section 2, we will use Poincaréseries to define, as in [5] and [13], harmonic weak Maass forms R (z) fk ∈ H2 k connected to the forms fk(z) Sk. Such forms may be written Rf (z) = Mf (z)+ − ∈ k k Nfk (z), where Mfk (z) is holomorphic on h and Nfk (z) is not holomorphic on h. We will find that the non-holomorphic contribution is a normalization of the period integral for fk(z),

Date: July 2, 2008. 2000 Mathematics Subject Classiﬁcation. 11F03, 11F11. The author thanks the National Science Foundation for its support through grant DMS-0600400.

1 2 MATTHEW BOYLAN given by i k 1 ∞ fk( τ) − (1.4) Nfk (z)= i(k 1)(2π) ck − 2 k dτ, − z ( i(τ + z)) − Z− − where (k 2)! (1.5) c := − R. k (4π)k 1 f (z) 2 ∈ − k k k Here, denotes the Petersson norm. We will also ﬁnd that the function Mfk (z), the holomorphick·k projection of R (z), has coeﬃcients c+ (n), and is given by fk fk

∞ + n 1 2k (1.6) Mf (z)= c (n)q =(k 1)!q− +(k 1)! + k fk − − B ··· n= 1 k X− See Theorem 2.1 for precise formulas for the coefficients c+ (n) as series with summands in fk terms of Kloosterman sums and values of modified Bessel functions of the first kind. The theory of harmonic weak Maass forms and mock theta functions is of great current interest. See, for example, the works [3], [4], [5], [6], [7], [8], [9], [13]. Let λ 1/2, 3/2 . In [3] and [6], Bringmann, Ono, and Folsom define a mock theta function to be a∈{ function which} arises as the holomorphic projection of a harmonic weak Maass form of weight 2 λ whose non-holomorphic part is a period integral of a linear combination of weight λ theta-series.− This definition encompasses the mock theta functions of Ramanujan, of which 2 ∞ qn f(q) := (1 + q)2(1 + q2)2 (1 + qn)2 n=1 X ··· is an important example. See [4] for its precise relation to Maass forms. In this sense, one may view the functions Mfk (z) as analogues of mock theta functions which we call mock modular forms. An important aspect of the study of mock modular forms concerns questions on the rationality (algebraicity) of their q-expansion coefficients. Mock modular forms arising from theta functions, such as f(q), are known to have rational (algebraic) q-expansion coefficients. In [8] and [9], the authors initiated a study of the algebraicity of coefficients of mock modular forms M(z)= c+(n)qn which do not arise from theta functions. In [8], Bruinier and Ono relate the algebraicity of the coefficients of certain weight 1/2 mock modular forms to the vanishing of derivativesP of quadratic twists of central critical values of weight two modular L-functions. In [9], Bruinier, Ono, and Rhoades give conditions which guarantee the algebraicity of coefficients of integer weight mock modular forms. In the proof of their result, they show that the coefficients c+(n) lie in the field Q(c+(1)). Part 1 of Corollary 1.3 below recovers this result for the coefficients c+ (n). However, the conditions in [9] guaranteeing fk the algebraicity of mock modular form coefficients do not apply in the present setting. As such, it is not known whether any c+ (1) is algebraic. fk + In [13], Ono conjectures, for all positive integers n, that the mock Delta coefficients, c∆(n), + are irrational. This conjecture has the following consequence. If c∆(1) (or any coefficient + c∆(n)) is irrational, then Lehmer’s Conjecture on the non-vanishing of the tau function is + true. Similarly, as a consequence of the work in this paper, we observe that if cfk (1) (or + any c (n)) is irrational, then for all positive integers n, we have af (n) = 0, which is the fk k 6 analogue of Lehmer’s conjecture for the forms fk. This is part 2 of Corollary 1.3 below. ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 3

In this paper, we use the Hecke operators and theta-operator to create certain weakly

holomorphic modular forms which are modifications of the mock modular forms Mfk (z). We then give explicit formulas which allow us to prove results on rationality and congruence properties of the coefficients of these weakly holomorphic forms. Our formulas also enable us to prove results on the values of these functions in the upper half-plane. To state our results, we require facts on weakly holomorphic modular forms on SL2(Z), forms which are holomorphic on h, but which may have a pole at the cusp infinity. We ! ! denote the space of such forms of integer weight k by Mk and note that Mk Mk. For more information on holomorphic and weakly holomorphic modular forms, see [12],⊆ for example. An important example of a weakly holomorphic form is the elliptic modular invariant,

3 ∞ E4(z) n 1 ! j(z) := = c(n)q = q− + 744 + 196884q + M . ∆(z) ···∈ 0 n= 1 X− We also require certain operators on q-expansions. If p is prime, then the pth Hecke operator in level one and integer weight k acts by

n k 1 n n (1.7) a(n)q T (p) := a(pn)+ p − a q , | k p X X where a n =0 if p ∤ n. For integers m 1, one defines Hecke operators T (m) in terms of p ≥ k the Hecke operators of prime index in (1.7). Furthermore, the Hecke operators preserve the ! spaces Mk and Mk. Next, we define the differential theta-operator 1 d d (1.8) θ := = q 2πi · dz dq and observe that θ a(n)qn = na(n)qn. X X Applying a normalization of the Hecke operators to j(z), we obtain an important sequence of modular forms. We set j0(z) := 1,

1 j (z) := j(z) 744 = q− + 196884q + , 1 − ··· and for all m 1, we set ≥

∞ m n ! (1.9) j (z) := m(j (z) T (m)) = q− + c (n)q M . m 1 | 0 m ∈ 0 n=1 X

The forms jm(z) have integer coeﬃcients and satisfy interesting properties. For example, let m τ h and deﬁne H (z) := ∞ j (τ)q . We will need the fact, proved in [2], that H (z) ∈ τ m=0 m τ is a weight two meromorphic modular form on SL2(Z) and that P ∞ θ(j(z) j(τ)) E (z) 1 (1.10) H (z)= j (τ)qm = − = 14 . τ m − j(z) j(τ) ∆(z) · j(z) j(τ) m=0 X − − 4 MATTHEW BOYLAN

To state our main result, we require further definitions. Let p be prime. The first function we study in connection with Mfk (z) is Lfk,p(z), defined by k 1 p − 1 k Lf ,p(z):= (Mf (z) T2 k(p) p − af (p)Mf (z)) k (k 1)! k | − − k k − ∞ 1 k 1 + + + n n (1.11) = p − c (pn) af (p)c (n)+ c q . (k 1)! fk − k fk fk p n= p − X− n ∞ We then define aLf ,p (n) by n= p aLf ,p (n)q := Lfk,p(z). To study the functions Lfk,p(z), k − k we introduce auxiliary functions. For k 12, 16, 18, 20, 22, 26 , let Fk be any form in Mk 2 and define P ∈{ } −

∞ n (1.12) Fk(z) := aFk (n)q . n=0 X For all positive integers t, deﬁne

t 2k (1.13) AF ,t(z) := aF (m)jt m(z)+ aF (0) σk 1(t) k k − k B − m=0 k X with jn(z) as in (1.9).

The second function we study in connection with Mfk (z) is

∞ k 1 1 k 1 + n (1.14) θ − (Mf (z)) = (k 1)!q− + n − c (n)q . k − − fk n=1 X Its study also requires the introduction of auxiliary functions. Let k be the least positive residue of k modulo 12. We deﬁne Gk(z) by

E12−k(z) k−k if k 10 (mod 12) ∆(z) 12 +1 6≡ (1.15) G (z) :=  k   E14(z)  k−10 if k 10 (mod 12). ∆(z) 12 +2 ≡  !  We note that Gk(z) M k Q((q)), and define aGk (n) C and nk Z by ∈ − ∩ ∈ ∈ ∞ n n (1.16) a (n)q := G (z)= q− k + Gk k ··· n= n X− k For fixed k and integers t n (with n as in (1.16)), we define ≤ k k nk

(1.17) BG ,t(z) := aG ( n)jn t(z) k k − − n=t X with jn(z) as in (1.9). k 1 Our main result gives exact formulas for Lfk,p(z) and θ − (Mfk (z)). Theorem 1.1. Let k 12, 16, 18, 20, 22, 26 , and let p be prime. ∈{ } ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 5

(1) In the notation above ((1.3), (1.11), (1.12), (1.13)), we have

AFk,p(z) afk (p)AFk,1(z) ! Lfk,p(z)= − M2 k. Fk(z) ∈ − (2) In the notation above ((1.6), (1.14), (1.16), (1.17)), we have

nk k 1 + m − cf (m)BGk,m(z) (k 1)!BGk, 1(z) k 1 m=1 k − ! θ − (Mfk (z)) = − − Mk. P Gk(z) ∈

Before proceeding, we illustrate Theorem 1.1 by applying it to f12(z) = ∆(z). Let p be prime. With F12(z)= E10(z) in part 1 of the theorem, we find that p 65520 247944 jp(z) 264 σ9(m)jp m(z) σ11(p) τ(p) j1(z) L (z)= − m=1 − − 691 − − 691 . ∆,p E (z) P 10 This is the main result in [13]. Using part 2 of the theorem, we obtain (1.18) θ11(M (z)) = ∆(z)(c+ (1) 11!(j (z)+24j (z) + 324)). ∆ ∆ − 2 1 On examining the right-hand sides in Theorem 1.1, we deduce properties of the forms k 1 Lfk,p(z) and θ − (Mfk (z)). First, we find that the coefficients of Lfk,p(z) are integers. Corollary 1.2. Let k 12, 16, 18, 20, 22, 26 , and let p be prime. Then for all integers n, we have ∈{ } 1 k 1 + + + n aL (n)= p − c (pn) af (p)c (n)+ c Z. fk,p (k 1)! fk − k fk fk p ∈ − A consequence of Corollary 1.2 is the following. Corollary 1.3. Let k 12, 16, 18, 20, 22, 26 . ∈{ } (1) For all integers n, we have c+ (n) Q(c+ (1)). fk fk + + ∈ (2) If c (1) (or any c (n)) is irrational, then for all integers n, we have af (n) =0. fk fk k 6 As an example of part 1, for all primes p, Corollary 1.2 gives

(k 1)! aLf ,p (1) a (p) (1.19) c+ (p)= − · k + fk c+ (1). fk k 1 k 1 fk p − p − · The general case follows by induction. As remarked above, the conclusion of part 2 of the corollary is the analogue of Lehmer’s conjecture for the forms fk. It is proved in Section 3. Theorem 1.1 and Corollary 1.2 permit us to apply well-known congruence properties for coeﬃcients of the Eisenstein series Ek(z) and the cusp forms fk(z) to obtain congruences for

Lfk,p(z).

Corollary 1.4. Let p be prime, and let τ(p)= af12 (p) as in (1.1). (1) If k 12, 16, 18, 20, 22, 26 , then we have ∈{ } j (z) (mod 24) if p =2 or p 23 (mod 24) p ≡ Lf ,p(z) j (z)+12j (z) (mod 24) if p =3 or p 11 (mod 24) k ≡  p 1 ≡ jp(z) (p + 1)j1(z) (mod 24) if p 5. − ≥  6 MATTHEW BOYLAN

(2) If k 18, 22, 26 , then we have ∈{ } j (z) 2j (z) (mod5) if p 1 (mod5) p − 1 ≡ Lf ,p(z) j (z)+(p + 1)(j (z)+2) (mod5) if p 2, 3 (mod5) k ≡  p 1 ≡ jp(z) (mod5) if p 4 (mod5). ≡ (3) If k 20, 26, then we have ∈{ } j (z) 2j (z) (mod7) if p 1 (mod7) p − 1 ≡ jp(z)+2(p + 1)(j1(z)+1) (mod7) if p 2, 4 (mod7) Lfk,p(z) ≡ ≡ jp(z)+3(p +1) (mod7) if p 3, 5 (mod7)  ≡ j (z) (mod7) if p 6 (mod7). p ≡  (4) If k 12, 22, then we have ∈{ } L (z) j (z) τ(p)j (z)+2(p +1 τ(p)) (mod 11). fk,p ≡ p − 1 − (5) We also have L (z) j (z) pτ(p)j (z) 2(p +1 pτ(p)) (mod 13) f26,p ≡ p − 1 − − L (z) j (z) a (p)j (z)+7(p +1 a (p)) (mod 17) f18,p ≡ p − f18 1 − f18 L (z) j (z) a (p)j (z)+5(p +1 a (p)) (mod 19). f20,p ≡ p − f20 1 − f20 Remark. In [13], Ono proved part 1 of Theorem 1.1 and Corollaries 1.2, 1.3, and 1.4 for f12(z) = ∆(z) S12. We also remark that Guerzhoy has proved related congruences. See Theorem 2 and∈ Corollary 1 of [10].

k 1 Next, we study values of Lfk,p(z) and θ − (Mfk (z)) at points in the upper half-plane. If g(z) is a function on h and τ h, then we define v (g(z)) to be the order of vanishing of g(z) ∈ τ at τ. Recalling (1.2), (1.10), and (1.12), we define a meromorphic modular form on SL2(Z) of weight k with coefficients b (n) C by Fk,τ ∈ ∞ b (n)qn := H (z)F (z) a (0)E (z). Fk,τ τ k − Fk k n=0 X Similarly, recalling (1.10) and (1.15), we define a meromorphic modular form on SL2(Z) of weight 2 k with coefficients β (n) C by − Gk,τ ∈ ∞ n βGk,τ (n)q := Hτ (z)Gk(z). n= n X− k Corollary 1.5. Let k 12, 16, 18, 20, 22, 26 , let τ h, and let p be prime. ∈{ } ∈ (1) We have

bFk,τ (p) afk (p)bFk,τ (1) Lfk,p(τ)= − Fk(τ) and nk k 1 + m − c (m)β ( m) (k 1)!β (1) k 1 m=1 fk Gk,τ Gk,τ θ − (Mfk (τ)) = − − − . P Gk(τ) ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 7

1 √ 3 (2) If ω = + − , then we have − 2 2 2 if k 0 (mod6) v (L (z)) ≡ ω fk,p ≥ 1 if k 4 (mod6) ( ≡ and

k 1 2 if k 2 (mod6) v (θ − (M (z))) ≡ ω fk ≥ 1 if k 4 (mod6). ( ≡ (3) We have v (L (z)) 1 if k 0 (mod4) i fk,p ≥ ≡ and k 1 v (θ − (M (z))) 1 if k 2 (mod4). i fk ≥ ≡ Remark. Setting Fk(z)= Ek 2(z) in Theorem 1.1, part 2 of the corollary implies, for k 0, 4 (mod 6), that − ≡

2(k 2) p 2k jp(ω) − m=1 σk 3(m)jp m(ω)+ σk 1(p) a (p)= − Bk−2 − − Bk − . fk 2(k 2) 2k P 744 − + − − Bk−2 Bk Similarly, part 3 of the corollary implies, for k 0 (mod 4), that ≡ 2(k 2) p 2k jp(i) − m=1 σk 3(m)jp m(i)+ σk 1(p) a (p)= − Bk−2 − − Bk − . fk 2(k 2) 2k P984 − + − Bk−2 Bk Guerzhoy proved these identities independently in [10] (Theorem 3). Corollary 1.3 underscores the significance of the coefficients c+ (1). Using Corollary 1.5, fk we provide alternative expressions for these coefficients. Perhaps they might shed some light on questions of rationality and algebraicity. Corollary 1.6. The following are true. ω if k 2, 4 (mod 6) (1) Set τ := ≡ . Let n be as in (1.16). Then we have i if k 2 (mod 4). k ( ≡ nk k 1 + (k 1)!BGk, 1(τ) m − cf (m)BGk,m(τ) c+ (1) = − − − m=2 k . fk B (τ) PGk,1 (2) There is a τ h with v (θ11(M (z)) 1. For such τ, we have ∈ τ ∆ ≥ + c∆(1) = 11!(j2(τ)+24j1(τ) + 324). Part 1 of the corollary follows from Theorem 1.1 and parts 2 and 3 of Corollary 1.5. Part 2 is proved in Section 3. The plan for Sections 2 and 3 of the paper is as follows. In Section 2.1, we provide the necessary background on harmonic weak Maass forms. In Section 2.2, we construct certain

Poincar´eseries and use them to deﬁne the functions Mfk (z). In Section 3, we prove Theorem 1.1 and its corollaries. 8 MATTHEW BOYLAN

2. Harmonic weak Maass forms and Poincare´ Series. 2.1. Harmonic weak Maass forms. We recall the deﬁnition of integer weight harmonic weak Maass forms on SL2(Z). Let k Z, let x, y R, and let z = x + iy h. The weight k hyperbolic Laplacian is deﬁned by ∈ ∈ ∈ ∂2 ∂2 ∂ ∂ ∆ := y2 + + iky + i . k − ∂x2 ∂y2 ∂x ∂y A harmonic weak Maass form of weight k on SL2(Z) is a smooth function f on h satisfying the following properties. a b (1) For all A = SL (Z) and all z h, we have c d ∈ 2 ∈ f(Az)=(cz + d)kf(z).

(2) We have ∆kf = 0. + n 1 εy (3) There is a polynomial Pf = n 0 cf (n)q C[[q− ]] such that f(z) Pf (z)= O(e− ) as y for some ε> 0. ≤ ∈ − →∞ P Using these conditions, one can show that harmonic weak Maass forms have q-expansions of the form

∞ ∞ + n n (2.1) f(z)= c (n)q + c−(n)Γ(k 1, 4πny)q− , f f − n=n0 n=1 X X where n Z and the incomplete Gamma function is given by 0 ∈ ∞ t a dt Γ(a, x) := e− t . t Zx For more information on this and other special functions used in this section, see, for example, [1]. We refer to the ﬁrst (respectively, second) sum in (2.1) as the holomorphic (respectively, non-holomorphic) projection of f(z). We also observe that M ! H and that the Hecke k ⊆ k operators preserve Hk.

2.2. PoincaréSeries. We now construct Poincaréseries as in [5]. Let k be an integer, let a b z h, and let A = SL (Z). For functions f : h C, we define ∈ c d ∈ 2 → k (f A)(z):=(cz + d)− f(Az). |k + α Now, let m Z, and let ϕm : R C be a function which satisfies ϕm(y)= O(y ) as y 0 for some α ∈ R. If β R, we set →e(β) := e2πiβ. In this notation, we define → ∈ ∈ ϕm∗ (z) := ϕm(y)e(mx). 1 n Noting that ϕm∗ (z) is invariant under the action of Γ := : n Z , we define ∞ ± 0 1 ∈ the Poincaréseries P (m,k,ϕ ; z) := (ϕ∗ A)(z). m m |k A Γ∞ SL2(Z) ∈ X\ We will study specializations of this series. ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 9

We ﬁrst consider the Poincar´eseries P (1, k, 1; z). It is well-known that P (1, k, 1; z) Sk. Since S has dimension one for k 12, 16, 18, 20, 22, 26 , the form P (1, k, 1; z) is a constant∈ k ∈{ } multiple of fk(z). Moreover, we observe that

(2.2) P (1, k, 1; z)= ckfk(z), with c as in (1.5). These facts may be found, for example, in 3.3 of [11]. k § Next, we consider certain non-holomorphic Poincaréseries. Let Mν,µ(z) be the M-Whittaker function. For k Z and s C, define ∈ ∈ k 2 s(y) := y − M k sgn(y),s 1 ( y ), M | | 2 − 2 | | and for integers m 1, set ϕ m(z) := 1 k ( 4πmy). Then, for integers k > 2, we define ≥ − M − 2 − the Poincaréseries Rfk (z) := P ( 1, 2 k,ϕ 1; z). Recalling the definition (1.4) of Nfk (z) and using (2.2), we have − − − i i k 1 ∞ fk( τ) k 1 ∞ P (1, k, 1, τ) − − Nfk (z)= i(k 1)(2π) ck − 2 k dτ = i(k 1)(2π) −2 k dτ. − z ( i(τ + z)) − − z ( i(τ + z)) − Z− − Z − We now define the series Mfk (z) by M (z) := R (z) N (z). fk fk − fk We require further special functions. For j Z, we denote by Ij the Ij-Bessel function. For m, n, c Z with c> 0, the Kloosterman sum∈ K(m, n, c) is given by ∈ mv + nv (2.3) K(m, n, c) := e , c v (Z/cZ)∗ ∈ X 1 where v is v− (mod c). We now state the main theorem of this section. It is a special case of Theorem 1.1 of [5]. It asserts that Mfk (z) is a mock modular form for fk(z) and provides formulas for its coefficients and the coefficients of Nfk (z). Theorem 2.1. Let k 12, 16, 18, 20, 22, 26 . ∈{ } (1) In the notation above, we have Rf (z)= Mf (z)+ Nf (z) H2 k. k k k ∈ − (2) The function Mfk (z) is holomorphic with q-series expansion

∞ 1 2k + n Mf (z)=(k 1)!q− +(k 1)! + c (n)q , k − − B fk k n=1 X where for integers n 1, we have ≥ k−1 ∞ K( 1,n,c) 4π√n + k 2 cf (n)= 2πi n− − Ik 1 . k − c − c c=1 X (3) The function Nfk (z) is non-holomorphic with q-series expansion

∞ m Nf (z)= c− (m)Γ(k 1, 4πmy)q− , k fk − m=1 X where for integers m 1, we have ≥ afk (m) c− (m)= (k 1)c , fk k k 1 − − m − with a (m) Z as in (1.3) and c R as in (1.5). fk ∈ k ∈ 10 MATTHEW BOYLAN

Remark 1. The functions Mfk (z) depend on the representation of fk(z) as a linear combination of Poincar´eseries P (m, k, 1; z) Sk. In this work, we choose the simplest representation, given by (2.2). ∈

Remark 2. We compute the constant term in the q-expansion of Mfk (z) using (2.3) and facts from elementary number theory. In [5], we ﬁnd that the constant term is

∞ + k K2 k( 1, 0,c) c (0) = (2πi) − − . fk − ck c=1 X We compute that mv mv K2 k( 1, 0,c)= e = e = µ(c), − − − c c v (Z/cZ)∗ v (Z/cZ)∗ ∈ X ∈ X where µ is the M¨obius function. Thus, since k is even, we have

∞ K2 k( 1, 0,c) ∞ µ(c) 1 k! − − = = = 2 , ck ck ζ(k) − · (2πi)kB c=1 c=1 k X X where ζ is the Riemann zeta-function.

Remark 3. We obtain the expansion for Nfk (z) from (1.4) as in the proof of Theorem 1.1 of [5]. 3. Proofs of Theorem 1.1 and its corollaries. We now turn to the proofs of Theorem 1.1 and its corollaries. The basic idea is as follows.

We use the Hecke operators and theta operator to eliminate Nfk (z), the non-holomorphic

part of the harmonic weak Maass form Rfk (z). This is where we use the fact that the forms fk lie in 1-dimensional spaces. What remains is a modiﬁcation of the mock modular form

Mfk (z) which is a weakly holomorphic modular form with an explicit principal part. Since SL2(Z) has genus one and only one cusp, one can express the weakly holomorphic modular form explicitly in terms of certain polynomials in j (Faber polynomials). To begin, we note that for all integers m and for k 12, 16, 18, 20, 22, 26 , the forms ∈ { } fk(z) are normalized eigenforms for the Hecke operators Tk(m) since the dimension of Sk is one. Therefore, for all primes p and for all integers n 1, from (1.7) we have ≥ k 1 n (3.1) a (pn)+ p − a = a (p)a (n). fk fk p fk fk

Next, we compute Nfk (z) T2 k(p). To do so, we recall the deﬁnitions of the U(p) and V (p) operators on functions f| : h− C: → p 1 1 − z + a (3.2) f(z) U(p) := f , | p p a=0 X (3.3) f(z) V (p) := f(pz). | We also recall, for all integers j, that the Hecke operators Tj(p) may be written as j 1 (3.4) f(z) T (p)= f(z) U(p)+ p − f(z) V (p). | j | | We now state two important facts. Though the ﬁrst was proved in [8] for half-integral weights, (see Theorem 7.4), we provide a short proof in our setting. ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 11

Proposition 3.1. Let p be prime, and let k 12, 16, 18, 20, 22, 26 . Then we have ∈{ } 1 k Nf (z) T2 k(p)= p − af (p)Nf (z), k | − k k

with afk (n) as in (1.3). Proof. Using part 3 of Theorem 2.1, (3.1), (3.2), (3.3), and (3.4), we compute:

1 k Nf (z) T2 k(p)= Nf (z) U2 k(p)+ p − Nf (z) V2 k(p) k | − k | − k | − ∞ ∞ afk (pn) n 1 k afk (n) pn = (k 1)c Γ(k 1, 4πny)q− + p − Γ(k 1, 4πpny)q− − − k (pn)k 1 − nk 1 − n=1 − n=1 − ! X X ∞ 1 k k 1 n Γ(k 1, 4πny) n = (k 1)c p − a (pn)+ p − a − q− − − k fk fk p nk 1 n=1 − X ∞ 1 k afk (n) n = (k 1)c p − a (p) Γ(k 1, 4πny)q− . − − k fk nk 1 − n=1 − X

The second fact we require is a special case of a result of Bruinier, Ono, and Rhoades [9] which follows from Bol’s identity relating the theta-operator and the Maass raising operator.

Theorem 3.2. If k 2 is an integer and f(z) H2 k has holomorphic projection M(z)= c+(n)qn, then we≥ have ∈ −

k 1 k 1 k 1 + n ! P θ − (f(z)) = θ − (M(z)) = n − c (n)q M . ∈ k X k 1 Remark. We emphasize that in [9], the authors precisely compute the image of θ − on spaces of harmonic weak Maass forms of weight 2 k < 0 on subgroups of type Γ0(N) with nebentypus character. − 3.1. Proof of Theorem 1.1. We may now prove Theorem 1.1. From part 1 of Theorem 2.1 and the fact that the Hecke operators preserve H2 k, we see that − k 1 p − 1 k Lf ,p(z) := (Rf (z) T2 k(p) p − af (p)Rf (z)) H2 k. k (k 1)! k | − − k k ∈ − − Then, by Proposition 3.1, part 2 of Theorem 2.1, and (1.7), we ﬁnd that

k 1 p − 1 k Lf ,p(z)= (Mf (z) T2 k(p) p − af (p)Mf (z)) k (k 1)! k | − − k k − p 1 2k = q− afk (p)q− + (σk 1(p) afk (p)) − Bk − − ∞ 1 k 1 + + + n n (3.5) + p − c (pn) af (p)c (n)+ c q . (k 1)! fk − k fk fk p n=1 − X ! Moreover, we observe that Lfk,p(z) M2 k is a weakly holomorphic modular form since it is a harmonic weak Maass form whose∈ holomorphic− projection is zero. 12 MATTHEW BOYLAN

n ! ∞ With Fk(z) = n=0 aFk (n)q Mk 2, we note that Lfk,p(z)Fk(z) M0. Using (3.5), we compute ∈ − ∈ P p m p 2k Lf ,p(z)Fk(z)= aF (m)q − + aF (0) σk 1(p) k k k B − m=0 k X 1 2k a (p) a (0)q− + a (1) + a (0) + O(q). − fk Fk Fk Fk B k

If we define AFk,t(z) as in (1.13), we find that L (z)F (z) (A (z) a (p)A (z)) = O(q) fk,p k − Fk,p − fk Fk,1 is a modular form of weight zero which is holomorphic on h and at infinity. Hence, it is constant. This constant is zero, thus completing the proof of part 1 of Theorem 1.1. ! Part 2 follows similarly. With Gk(z) M k as in (1.15), we see from Theorem 3.2 that k 1 ! ∈ − θ − (M (z))G (z) M . Using (1.14) and (1.16), we compute fk k ∈ 0 nk nk nk k 1 n 1 k 1 + m n θ − (Mf (z))Gk(z)= (k 1)! aG ( n)q− − + m − c (m) aG ( n)q − + O(q). k − − k − fk k − n= 1 m=1 n=m X− X X

If we deﬁne BGk,t(z) as in (1.17), we ﬁnd that

nk k 1 k 1 + θ − (Mfk (z))Gk(z) m − cf (m)BGk,m(z) (k 1)!BGk, 1(z) = O(q) − k − − − m=1 ! X is a modular form of weight zero which is holomorphic on h and at inﬁnity. The conclusion follows.

3.2. Proof of Corollaries 1.2 and 1.3. To prove Corollary 1.2, it suﬃces to show that L (z) Z((q)). We consider cases. fk,p ∈ When k 12, 16 , we apply Theorem 1.1 with Fk(z) = Ek 2(z) Mk 2 Z[[q]]. Here, ∈ {2(k 2) } − 691 ∈ − ∩3617 we note that − Z. Furthermore, we observe that B12 = , B16 = , and that Bk−2 ∈ − 2730 − 510 for all primes p, σ (p) a (p)= τ(p) (mod 691) 11 ≡ f12 σ (p) a (p) (mod 3617). 15 ≡ f16 These congruences may be found, for example, in [14]. Therefore, for k 12, 16 , we must have ∈{ } 2k (3.6) (σk 1(p) afk (p)) Z. Bk − − ∈ Hence, from (1.2), (1.13), (3.6), and Theorem 1.1, we deduce that

AEk−2,p(z) afk (p)AEk−2,1(z) (3.7) Lfk,p(z)= − Ek 2(z) − 2(k 2) p 2(k 2) 2k j (z) − σ (m)j (z) a (p) j (z) − + (σ (p) a (p)) p B −2 m=1 k 3 p m fk 1 B −2 B k 1 fk = − k − − − − k k − − Ek 2(z) P − has integer coeﬃcients. ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 13

When k 18, 20, 22 , we apply Theorem 1.1 with Fk(z) = ∆(z)Ek 14(z) Sk Z[[q]], from which∈ we { obtain } − ∈ ∩ p aF (m)jp m(z) af (p) m=1 k − k Lfk,p(z)= − Z((q)). ∆(z)Ek 14(z) ∈ P − 2 Lastly, when k = 26, we apply Theorem 1.1 with F26(z) =∆(z) S24 Z[[q], which gives p ∈ ∩ aF (m)jp m(z) L (z)= m=2 k − Z((q)). f26,p ∆(z)2 ∈ P

To see part 2 of Corollary 1.3, we ﬁrst remark that if afk (p) = 0 for all primes p, then + 6 af (n) = 0 for all n. Therefore, we suppose that c (m) is irrational for some m and that k 6 fk afk (p) = 0 for some prime p. Corollary 1.2 implies that for all n,

1 k 1 + + n p − c (pn)+ c Z. (k 1)! · fk fk p ∈ − Setting n = 1, we see that c+ (p) Q. But then (1.19) shows that c+ (1) Q. Part 1 of the fk ∈ fk ∈ corollary now implies that c+ (m) Q(c+ (1)) must be rational, a contradiction. fk ∈ fk 3.3. Proof of Corollary 1.4. On applying Theorem 1.1 with Fk(z) = Ek 2(z) Mk 2, − ∈ − we obtain Lfk,p(z) as in (3.7). We then use the von Staudt-Claussen congruences for the denominators of Bernoulli numbers (see [12], Lemma 1.22, for example) and congruences for

the coeﬃcients afk (p) (see [14], Corollary to Theorem 4, for example) to prove Corollary 1.4. 2j To begin, we note that for all j 2 and even, we have 0 (mod 24), so Ej(z) 1 ≥ Bj ≡ ≡ (mod 24). Moreover, for k 12, 16, 18, 20, 22, 26 , we have fk(z) ∆(z) (mod 24). Part 1 of the corollary holds since∈{ for all primes p, we have} ≡ 0 (mod24) if p =2 τ(p)= a (p) 12 (mod24) if p =3 f12 ≡  p +1 (mod24) if p 5. ≥ 2(k 2) Next, we note that if ℓ is prime and ℓ 1 k 2, then we have − 0 (mod ℓ), so − | − Bk−2 ≡ Ek 2(z) 1 (mod ℓ). From (3.7), we see that if ℓ 1 k 2, then − ≡ − | − 2k (3.8) Lfk,p(z) jp(z) afk (p)j1(z)+ (p +1 afk (p)) (mod ℓ). ≡ − Bk − When ℓ 5, this is the case for pairs ≥ (ℓ,k) (5, 18), (5, 22), (5, 26), (7, 20), (7, 26), (11, 12), (11, 22), (13, 26), (17, 18), (19, 20) . ∈{ } The remaining parts of Corollary 1.4 are specializations of (3.8) using congruences for fk(z), with θ as in (1.8), and congruences for τ(p) when p is prime. These congruences are: θ∆(z) (mod5) if k 18, 22, 26 ∈{ } θ∆(z) (mod7) if k 20, 26 (3.9) fk(z)  ∈{ } ≡ ∆(z) (mod11) if k 12, 22  ∈{ } θ∆(z) (mod13) if k = 26,

 2 p + p (mod 5) (3.10) τ(p) 4 ≡ (p + p (mod 7). 14 MATTHEW BOYLAN

To prove the congruences for fk(z) modulo 5, 7, and 13, we note that θ∆(z) = ∆(z)E2(z) and that if k k′ (mod ℓ 1), then Kummer’s congruences for Bernoulli numbers imply ≡ − that Ek(z) Ek′ (z) (mod ℓ). For k ≡18, 22, 26 , (3.9) and (3.10) imply that ∈{ } p a (p) pτ(p) p(p + p2) (p +1) (mod5), fk ≡ ≡ ≡ 5 2k where · is the Legendre symbol. Using that 1 (mod 5), we obtain part 2 of the Bk ≡ corollary.· For k 20, 26 , (3.9) and (3.10) imply that ∈{ } p a (p) pτ(p) p(p + p4) p2 1+ (mod 7). fk ≡ ≡ ≡ 7 Using the fact that 2k 3 (mod 7) gives part 3 of the corollary. For k 12, 22 , (3.9) Bk ≡ ∈ { } together with the fact that 2k 2 (mod 11) gives part 4 of the corollary. Part 5 of the Bk ≡ corollary is similar.

3.4. Proof of Corollaries 1.5 and 1.6. Recalling (1.2), (1.10), (1.12), and (1.13), for n 1, we find that the nth coefficient of ≥ ∞ b (n)qn = H (z)F (z) a (0)E (z) Fk,τ τ k − Fk k n=0 X is given by n 2k bF ,τ (n)= aF (m)jn m(τ)+ aF (0) σk 1(n)= AF ,n(z). k k − k B − k m=0 k X Similarly, from (1.10), (1.16), and (1.17), we find that the nth coefficient of

∞ n βGk,τ (n)q = Hτ (z)Gk(z) n= n X− k is given by

n nk

βG ,τ (n)= aG (m)jn m(τ)= aG ( m)jm+n(τ)= BG , n(τ). k k − k − k − m= n m= n X− k X− Theorem 1.1 now implies part 1 of Corollary 1.5. Next, noting from (3.5) that p ! Lfk,p(z)= q− + M2 k, ···∈ − we ﬁnd that p p ∆(z) (AFk,p(z) afk (p)AFk,1(z)) Lfk,p(z)∆(z) = − =1+ M12p+2 k. Fk(z) ···∈ −

Applying the valence formula for forms in M12p+2 k (see [12], Theorem 1.29), we see that − v (L (z)) 2(k +1) (mod 3), ω fk,p ≡ k v (L (z)) +1 (mod2). i fk,p ≡ 2 ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 15

Since these quantities are non-negative (Lfk,p (z) is weakly holomorphic), we obtain parts 2 and 3 of the corollary. Similarly, observing from (1.14) that k 1 θ − (M (z))∆(z)= (k 1)! + M , fk − − ···∈ k+12 we apply the valence formula for forms in Mk+12 to deduce parts 2 and 3 of the corollary for k 1 the weakly holomorphic modular form θ − (Mfk (z)). To obtain part 2 of Corollary 1.6, we first note that for all τ h we have vτ (∆(z)) = 0. Then applying the valence formula to ∈ θ11(M (z)) ∆(z)= 11! + M ∆ · − ···∈ 24 gives 1 1 v (θ11(M (z)) + v (θ11(M (z)) + v (θ11(M (z))=2, 2 · i ∆ 3 · ω ∆ P ∆ P where the sum is over all P h inequivalent to i, Xω in a fundamental domain. Since 11 ∈ 11 θ (M∆(z)) is weakly holomorphic, for all τ h, we have vτ (θ (M∆(z)) 0. Hence, for some τ h, we must have v (θ11(M (z)) 1.∈ Since ∆(τ) = 0, (1.18) implies≥ the result. ∈ τ ∆ ≥ 6 Acknowledgment The author thanks K. Ono for helpful comments and suggestions in the preparation of this paper. References [1] M. Abramowitz and I. Stegun, Handbook of mathematical functions. Dover Publications, 1965. [2] T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain modular functions and an application. Comm. Math. Univ. Sancti Pauli 46 (1997), 93-101. [3] K. Bringmann, A. Folsom, and K. Ono, q-series and weight 3/2 Maass forms. Preprint. [4] K. Bringmann and K. Ono, The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165 (2006), 243-266. [5] K. Bringmann and K. Ono, Lifting elliptic cusp forms to Maass forms with an application to partitions. Proc. Natl. Acad. Sci. USA 104 (2007), no. 10, 3725-3731. [6] K. Bringmann and K. Ono, Dyson’s ranks and Maass forms. Ann. of Math., to appear. [7] K. Bringmann, K. Ono, and R.C. Rhoades, Eulerian series as modular forms. J. Amer. Math. Soc., to appear. [8] J. H. Bruinier and K. Ono, Heegner divisors, L-functions, and harmonic weak Maass forms. Preprint. [9] J. H. Bruinier, K. Ono, and R.C. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues. Math. Ann., to appear. [10] P. Guerzhoy, Hecke operators for weakly holomorphic modular forms and supersingular congruences. Preprint. [11] H. Iwaniec, Topics in the classical theory of automorphic forms. Grad. Stud. in Math. 17, Amer. Math. Soc., Providence, RI, 1997. [12] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS 102, Amer. Math. Soc., 2004. [13] K. Ono, A mock theta function for the Delta-function. Proceedings of the 2007 Integers Conference, to appear. [14] H. P. F. Swinnerton-Dyer, On ℓ-adic representations and congruences for coefficients of modular forms. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math., Vol. 350, Springer Berlin, 1973, 1-55.

Department of Mathematics, University of South Carolina, Columbia, SC 29208 E-mail address: [email protected]